A good way of illustrating this is that John doesn’t even attempt a nativity story. He starts “In the beginning was the Word, and the Word was with God, and the Word was God.”
The nativity was important for Jewish converts, partly because of descending from kind David (you'll notice the lengthy listings of ancestors in the synoptic gospels).
Yep, but more specifically Jesus as the Logos, which is a core concept in Platonism and Stoicism.
John very clearly meant his gospel to be read by gentiles from a Greek background.